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Unformatted text preview: E [ Y ] = E [ X Â· Y/X ] = E [ X ] E [ Y/X ] = l 2 1 2 = l 4 3. (a) Z 1 Z 1 1y cxy dxdy = c Â· 1 2 Z 1 y [1(1y ) 2 ] dy = c 2 Z 1 2 y 2y 3 dy = c 2 (2 Â· 1 31 4 ) = 5 c 24 = 1 Therefore, c = 24 5 (b) Z 3 4 1 2 Z 1 2 1y cxy dxdy = 53 1280 (c) Z 1 1x cxy dy = cx 1 2 (2 xx 2 ) = 12 5 (2 x 2x 3 ) 4. p X ( k ) = P ( X = k ) = Z âˆž P ( X = k  Î» = m ) f Î» ( m ) dm = Z âˆž m k k ! em em dm = Z âˆž m k k ! e2 m dm = 1 2 k +1 k = 0 , 1 , ... 5. (a) P ( X < Y ) = R 1 R 1 x f ( x, y ) dydx (b) P ( X + Y â‰¤ 1) = R 1 R 1x f ( x, y ) dydx (c) Since X and Y are continuous random variables, it is the same as that in (a). (d) P ( XY â‰¤ 1 / 2) = R 1 2 R 1 f ( x, y ) dydx + R 1 1 2 R 1 2 x f ( x, y ) dydx...
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This note was uploaded on 12/11/2011 for the course IEOR 3658 taught by Professor Olvera during the Fall '08 term at Columbia.
 Fall '08
 Olvera

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